Three integers have a sum of 7 and a product of 0. If the difference of the greatest number and the least number is 11, then the least of these numbers is?
If they have a product of 0, then at least one of the numbers is 0.
The greatest number cannot be 0, otherwise the sum would be negative.
Now, let x = the greatest number
x - 11 = the least number
x + x - 11 + 0 = 7
2x - 11 = 7
2x = 18
x = 9
The least number is 9-11, or -2.
If their product is zero, then one of them had to be a zero
so the 3 numbers can be called x,0, and y
x+y+0 = 7 or
x+y = 7 and
x-y = 11
adding them: 2x = 18
x = 9
if x=9 then y = -2
the 3 numbers are -2,0, and 9
check: is their sum = 7 ? YES
is the largest - the smallest = 11 ? YES
Let's assume the three integers as x, y, and z.
Given:
x + y + z = 7 ----(1)
x * y * z = 0 ----(2)
z - x = 11 ----(3)
From equation (2), we can conclude that at least one of the three integers, x, y, or z, must be equal to zero.
Case 1: If x = 0
From equation (1):
0 + y + z = 7
=> y + z = 7 ----(4)
From equation (3):
z - 0 = 11
=> z = 11 ----(5)
Substituting the value of z from equation (5) in equation (4):
y + 11 = 7
=> y = -4 ----(6)
In this case, the least of the three integers is -4.
Case 2: If y = 0
From equation (1):
x + 0 + z = 7
=> x + z = 7 ----(7)
From equation (3):
z - x = 11
We can rearrange equation (7) as:
x = 7 - z
Substituting the value of x in equation (3):
z - (7 - z) = 11
=> z - 7 + z = 11
=> 2z - 7 = 11
=> 2z = 11 + 7
=> 2z = 18
=> z = 9 ----(8)
Substituting the value of z from equation (8) in equation (7):
x + 9 = 7
=> x = -2 ----(9)
In this case, the least of the three integers is -2.
Case 3: If z = 0
From equation (1):
x + y + 0 = 7
=> x + y = 7 ----(10)
From equation (3):
0 - x = 11
=> x = -11 ----(11)
Substituting the value of x from equation (11) in equation (10):
-11 + y = 7
=> y = 18 ----(12)
In this case, the least of the three integers is -11.
Comparing the three cases, we find that the least of the three integers is -11.
Therefore, the least of these numbers is -11.
To solve this problem, we can set up a system of equations based on the given information.
Let's assume the three integers are a, b, and c, where a is the greatest number, b is the middle number, and c is the least number.
Given that their sum is 7, we can write the equation:
a + b + c = 7 .......(equation 1)
Given that their product is 0, we can write the equation:
abc = 0 .......(equation 2)
Given that the difference between the greatest and least number is 11, we can write the equation:
a - c = 11 .......(equation 3)
To find the least of the three numbers, we need to solve this system of equations.
First, let's use equation 2. Since the product of three numbers is 0, one of the three variables must be 0. Therefore, we have three possibilities:
1) a = 0, b ≠ 0, c ≠ 0
2) a ≠ 0, b = 0, c ≠ 0
3) a ≠ 0, b ≠ 0, c = 0
Let's examine each case and see if it satisfies the remaining equations:
Case 1: a = 0, b ≠ 0, c ≠ 0
From equation 1, we have: 0 + b + c = 7
This implies b + c = 7
From equation 3, we have: 0 - c = 11
This implies c = -11
Since c ≠ 0, this case does not satisfy the given conditions.
Case 2: a ≠ 0, b = 0, c ≠ 0
From equation 1, we have: a + 0 + c = 7
This implies a + c = 7
From equation 3, we have: a - c = 11
Adding the equations: (a + c) + (a - c) = 7 + 11
This simplifies to 2a = 18
So, a = 9
Since a ≠ 0, this case does not satisfy the given conditions.
Case 3: a ≠ 0, b ≠ 0, c = 0
From equation 1, we have: a + b + 0 = 7
This implies a + b = 7
From equation 3, we have: a - 0 = 11
This implies a = 11
Since a ≠ 0, this case does not satisfy the given conditions.
Therefore, none of the cases satisfy all the given conditions.
Hence, there are no three integers that satisfy all the given conditions.